The Ground State Energy of a Two-Dimensional Bose Gas

We prove the following formula for the ground state energy density of a dilute Bose gas with density $\rho$ in 2 dimensions in the thermodynamic limit $\begin{array}{c}{e}^{\text{2D}}\left(\rho \right)=4\pi {\rho }^{2}Y(1-Y|logY|+(2\Gamma +\frac{1}{2}+log\left(\pi \right))Y)+o\left({\rho }^2{Y}^2\right),\end{array}$ as $\rho a^2\to 0$ . Here $Y=|log\left(\rho a^2\right){|}^{-1}$ and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee-Huang-Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.